10,410 research outputs found
Operator-Theoretic Characterization of Eventually Monotone Systems
Monotone systems are dynamical systems whose solutions preserve a partial
order in the initial condition for all positive times. It stands to reason that
some systems may preserve a partial order only after some initial transient.
These systems are usually called eventually monotone. While monotone systems
have a characterization in terms of their vector fields (i.e. Kamke-Muller
condition), eventually monotone systems have not been characterized in such an
explicit manner. In order to provide a characterization, we drew inspiration
from the results for linear systems, where eventually monotone (positive)
systems are studied using the spectral properties of the system (i.e.
Perron-Frobenius property). In the case of nonlinear systems, this spectral
characterization is not straightforward, a fact that explains why the class of
eventually monotone systems has received little attention to date. In this
paper, we show that a spectral characterization of nonlinear eventually
monotone systems can be obtained through the Koopman operator framework. We
consider a number of biologically inspired examples to illustrate the potential
applicability of eventual monotonicity.Comment: 13 page
Generalised Compositional Theories and Diagrammatic Reasoning
This chapter provides an introduction to the use of diagrammatic language, or
perhaps more accurately, diagrammatic calculus, in quantum information and
quantum foundations. We illustrate the use of diagrammatic calculus in one
particular case, namely the study of complementarity and non-locality, two
fundamental concepts of quantum theory whose relationship we explore in later
part of this chapter.
The diagrammatic calculus that we are concerned with here is not merely an
illustrative tool, but it has both (i) a conceptual physical backbone, which
allows it to act as a foundation for diverse physical theories, and (ii) a
genuine mathematical underpinning, permitting one to relate it to standard
mathematical structures.Comment: To appear as a Springer book chapter chapter, edited by G.
Chirabella, R. Spekken
Entanglement Renormalization: an introduction
We present an elementary introduction to entanglement renormalization, a real
space renormalization group for quantum lattice systems. This manuscript
corresponds to a chapter of the book "Understanding Quantum Phase Transitions",
edited by Lincoln D. Carr (Taylor & Francis, Boca Raton, 2010)Comment: v2: new format. 24 pages, 10 figures, 2 tables, chapter of the book
"Understanding Quantum Phase Transitions", edited by Lincoln D. Carr (Taylor
& Francis, Boca Raton, 2010
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Scheme for constructing graphs associated with stabilizer quantum codes
We propose a systematic scheme for the construction of graphs associated with
binary stabilizer codes. The scheme is characterized by three main steps:
first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum
code; second, the canonical form of the CWS code is uncovered; third, the input
vertices are attached to the graphs. To check the effectiveness of the scheme,
we discuss several graphical constructions of various useful stabilizer codes
characterized by single and multi-qubit encoding operators. In particular, the
error-correcting capabilities of such quantum codes are verified in
graph-theoretic terms as originally advocated by Schlingemann and Werner.
Finally, possible generalizations of our scheme for the graphical construction
of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum
codes are briefly addressed.Comment: 42 pages, 12 figure
Scattering particles in quantum spin chains
A variational approach for constructing an effective particle description of
the low-energy physics of one-dimensional quantum spin chains is presented.
Based on the matrix product state formalism, we compute the one- and
two-particle excitations as eigenstates of the full microscopic Hamiltonian. We
interpret the excitations as particles on a strongly-correlated background with
non-trivial dispersion relations, spectral weights and two-particle S matrices.
Based on this information, we show how to describe a finite density of
excitations as an interacting gas of bosons, using its approximate
integrability at low densities. We apply our framework to the Heisenberg
antiferromagnetic ladder: we compute the elementary excitation spectrum and the
magnon-magnon S matrix, study the formation of bound states and determine both
static and dynamic properties of the magnetized ladder.Comment: published versio
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